Introduction to Quantum Statistics - math.leidenuniv.nlgill/Limerick2009-1.pdf · Introduction to...

Introductionto

Quantum Statistics

Richard Gill

Mathematical InstituteUniversity of Leiden

Seminar, Limerick, 2009

Collaborators

Madalin Guta (Nottingham)

Jonas Kahn (Lille)

Stefan Zohren (Leiden)

Wojtek Kotlowski (Amsterdam)

Cristina Butucea (Lille)

Peter Jupp (St. Andrews)

Ole E. Bardorff-Nielsen (Aarhus)

Slava Belavkin (Nottingham)

Plan of first lecture

Historical remarks and motivation

The basic notions: states, measurements, channels

Current topics in Quantum Statistics

State estimation; Quantum Cramer-Rao

Plan of second lecture

Local asymptotic normality for i.i.d. states

Quantum Cramer-Rao revisited

[Local asymptotic normality for quantum Markov chains]

Plan of third lecture

Quantum learning, sparsity

Bell inequalities, quantum non-locality

The measurement problem: the new eventum mechanics

Don’t worry...

There will be no third lecture!

Quantum mechanics up to the 60’s

Q.M. predicts probability distributions of measurement outcomes

Perform measurements on huge ensembles

Observed frequencies = probabilities

Old Paradigm

It makes no sense to talk about individual quantum systems

E. Schrodinger[“Are there quantum jumps ?”, British J.Phil. Science 1952]

“We are not experimenting with single particles, anymore than we can raise Ichtyosauria in the zoo.

We are scrutinizing records of events long after theyhave happened.”

Are there quantum jumps ?

Paul ion trap

This article reviews the various quantum-jump ap-proaches developed over the past few years. We focuson the theoretical description of basic dynamics and onsimple instructive examples rather than the applicationto numerical simulation methods.

Some of the topics covered here can also be found inearlier summaries (Erber et al., 1989; Cook, 1990;Mo” lmer and Castin, 1996; Srinivas, 1996) and more re-cent summer school lectures (Mo” lmer, 1994; Zoller andGardiner, 1995; Knight and Garraway, 1996).

II. INTERMITTENT FLUORESCENCE

Quantum mechanics is a statistical theory that makesprobabilistic predictions of the behavior of ensembles(ideally an infinite number of identically prepared quan-tum systems) using density operators. This descriptionwas completely sufficient for the first 60 years of theexistence of quantum mechanics because it was gener-ally regarded as completely impossible to observe andmanipulate single-quantum systems. For example,Schrodinger (1952) wrote. . . we never experiment with just one electron or atom or(small) molecule. In thought experiments we sometimesassume that we do; this invariably entails ridiculous con-sequences. . . . . In the first place it is fair to state that weare not experimenting with single particles, any more thanwe can raise Ichthyosauria in the zoo.

This (rather extreme) opinion was challenged by a re-markable idea of Dehmelt, which he first made public in1975 (Dehmelt, 1975, 1982). He considered the problemof high-precision spectroscopy, where one wants to mea-sure the transition frequency of an optical transition asaccurately as possible, e.g., by observing the resonancefluorescence from that transition as part (say) of anoptical-frequency standard. However, the accuracy ofsuch a measurement is fundamentally limited by thespectral width of the observed transition. The spectralwidth is due to spontaneous emission from the upperlevel of the transition, which leads to a finite lifetime t ofthe upper level. Basic Fourier considerations then implya spectral width of the scattered photons of the order oft21. To obtain a precise value of the transition fre-quency, it would therefore be advantageous to excite ametastable transition that scatters only a few photonswithin the measurement time. On the other hand, onethen has the problem of detecting these few photons,and this turns out to be practically impossible by directobservation. Dehmelt’s proposal, however, suggests asolution to these problems, provided one would be ableto observe and manipulate single ions or atoms, whichbecame possible with the invention of single-ion traps(Paul et al., 1958; Paul, 1990) (for a review, see Horvathet al., 1997). We illustrate Dehmelt’s idea in its originalsimplified rate-equation picture. It runs as follows.

Instead of observing the photons emitted on the meta-stable two-level system directly, he proposed using anoptical double-resonance scheme as depicted in Fig. 1.One laser drives the metastable 0↔2 transition while asecond strong laser saturates the strong 0↔1; the life-

time of the upper level 1 is, for example, 1028 s, whilethat of level 2 is of the order of 1 s. If the initial state ofthe system is the lower state 0, then the strong laser willstart to excite the system to the rapidly decaying level 1,which will then lead to the emission of a photon after atime that is usually very short (of the order of the life-time of level 1). This emission restores the system to thelower level 0; the strong laser can start to excite thesystem again to level 1, which will emit a photon on thestrong transition again. This procedure repeats until atsome random time the laser on the weak transition man-ages to excite the system into its metastable state 2,where it remains shelved for a long time, until it jumpsback to the ground state, either by spontaneous emissionor by stimulated emission due to the laser on the 0↔2transition. During the time the electron rests in themetastable state 2, no photons will be scattered on thestrong transition, and only when the electron jumps backto state 0 can the fluorescence on the strong transitionstart again. Therefore, from the switching on and off ofthe resonance fluorescence on the strong transition(which is easily observable), we can infer the extremelyrare transitions on the 0↔2 transition. Therefore wehave a method to monitor rare quantum jumps (transi-tions) on the metastable 0↔2 transition by observationof the fluorescence from the strong 0↔1 transition.

A typical experimental fluorescence signal is depictedin Fig. 2 (Thompson, 1996), where the fluorescence in-tensity I(t) is plotted. However, this scheme only worksif we observe a single quantum system, because if weobserve a large number of systems simultaneously, therandom nature of the transitions between levels 0 and 2implies that some systems will be able to scatter photonson the strong transition, while others will not becausethey are in their metastable state at that moment. Froma large collection of ions observed simultaneously, onewould then obtain a more or less constant intensity ofphotons emitted on the strong transition.

The calculation of this mean intensity is a straightfor-ward task using standard Bloch equations. The calcula-

FIG. 1. The V system. Two upper levels 1 and 2 couple to acommon ground state 0. The transition frequencies are as-sumed to be largely different so that each of the two lasersdriving the system couples to only one of the transitions. The1↔0 transition is assumed to be strong while the 2↔0 transi-tion is weak.

102 M. B. Plenio and P. L. Knight: Quantum-jump approach to dissipative dynamics . . .

Rev. Mod. Phys., Vol. 70, No. 1, January 1998

3 level atom driven by 2 lasers

tion of single-system properties, such as the distributionof the lengths of the periods of strong fluorescence, re-quired some effort, which eventually led to the develop-ment of the quantum-jump approach. Apart from theinteresting theoretical implications for the study of indi-vidual quantum systems, Dehmelt’s proposal obviouslyhas important practical applications. An often-cited ex-ample is the realization of a new time standard using asingle atom in a trap. The key idea here is to use eitherthe instantaneous intensity or the photon statistics of theemitted radiation on the strong transition (the statisticsof the bright and dark periods) to stabilize the frequencyof the laser on the weak transition. This is possible be-cause the photon statistics of the strong radiation de-pends on the detuning of the laser on the weak transi-tion (Kim, 1987; Kim and Knight, 1987; Kim et al., 1987;Ligare, 1988; Wilser, 1991). Therefore a change in thestatistics of bright and dark periods indicates that thefrequency of the weak laser has shifted and has to beadjusted. However, for continuously radiating lasers thisfrequency shift will also depend on the intensity of thelaser on the strong transition. Therefore, in practice,pulsed schemes are preferable for frequency standards(Arecchi et al., 1986; Bergquist et al., 1994).

Due to the inability of experimentalists to store, ma-nipulate, and observe single-quantum systems (ions) atthe time of Dehmelt’s proposal, both the practical andthe theoretical implications of his proposal were not im-mediately investigated. It was about ten years later thatthis situation changed. At that time Cook and Kimble(1985) made the first attempt to analyze the situationdescribed above theoretically. Their advance was stimu-lated by the fact that by that time it had become possibleto actually store single ions in an ion trap (Paul trap;Paul et al., 1958; Neuhauser et al., 1980; Paul, 1990).

In their simplified rate-equation approach Cook andKimble started with the rate equations for an incoher-ently driven three-level system as shown in Fig. 1 and

assumed that the strong 0↔1 transition is driven tosaturation. They consequently simplify their rate equa-tions, introducing the probabilities P1 of being in themetastable state and P2 of being in the strongly fluo-rescing 0↔1 transition. This simplification now allowsthe description of the resonance fluorescence to be re-duced to that of a two-state random telegraph process.Either the atomic population is in the levels 0 and 1, andtherefore the ion is strongly radiating (on), or the popu-lation rests in the metastable level 2, and no fluores-cence is observed (off). They then proceed to calculatethe distributions for the lengths of bright and dark peri-ods and find that their distribution is Poissonian. Theiranalysis, which we have outlined very briefly here, is ofcourse very much simplified in many respects. The mostimportant point is certainly the fact that Cook andKimble assume incoherent driving and therefore adopt arate-equation model. In a real experiment coherent ra-diation from lasers is used. The complications arising incoherent excitation finally led to the development of thequantum-jump approach. Despite these problems, theanalysis of Cook and Kimble showed the possibility ofdirect observation of quantum jumps in the fluorescenceof single ions, a prediction that was confirmed shortlyafterwards in a number of experiments (Bergquist et al.,1986; Nagourney et al., 1986a, 1986b; Sauter et al., 1986a,1986b; Dehmelt, 1987) and triggered a large number ofmore detailed investigations, starting with early worksby Javanainen (1986a, 1986b, 1986c). The subsequent ef-fort of a great number of physicists eventually culmi-nated in the development of the quantum-jump ap-proach. Before we present this development in greaterdetail, we should like to study in slightly more detailhow the dynamics of the system determines the statisticsof bright and dark periods. Again assume a three-levelsystem as shown in Fig. 1. Provided the 0↔1 and 0↔2Rabi frequencies are small compared with the decayrates, one finds for the population in the strongly fluo-rescing level 1 as a function of time something like thebehavior shown in Fig. 3 (we derive this in detail in alater section). We choose for this figure the valuesg1@g2 for the Einstein coefficients of levels 1 and 2,which reflects the metastability of level 2. For timesshort compared with the metastable lifetime g2

21 , theatomic dynamics can hardly be aware of level 2 andevolve as a 0–1 two-level system with the ‘‘steady-state’’population r 11 of the upper level. After a time g2

21 , themetastable state has an effect, and the (ensemble-averaged) population in level 1 reduces to the appropri-ate three-level equilibrium values. The ‘‘hump’’ Dr11shown in Fig. 3 is actually a signature of the telegraphicfluorescence discussed above. To show this, consider afew sequences of bright and dark periods in the tele-graph signal as shown in Fig. 4. The total rate of emis-sion R is proportional to the rate in a bright periodtimes the fraction of the evolution made up of brightperiods. This gives

R5g1 r 11S TL

TL1TDD , (1)

FIG. 2. Recorded resonance fluorescence signal exhibitingquantum jumps from a laser-excited 24Mg+ ion (Thompson,1996). Periods of high photon count rate are interrupted byperiods with negligible count rate (except for an unavoidabledark-count rate).

103M. B. Plenio and P. L. Knight: Quantum-jump approach to dissipative dynamics . . .


Recorded fluorescence signal from 1 ion

First experiments with individual quantum systems

Measurements with stochastic outcomes

Stochastic Schrodinger equations

Influx of mathematical ideas in the 70’s

E. B. Davies V. P. Belavkin A. S. Holevo C. W. Helstrom

Probability

what is the nature of quantum noise ?

Filtering Theory

what happens to the quantum system during measurement ?

Information Theory

how to encode, transmit and decode quantum information ?

Statistics

what do we learn from measurement outcomes ?

Quantum Information and Technology

New Paradigm

Individual quantum systems are carriers of a new type of information

Emerging fields:

Quantum Information Processing

Quantum Computation and Cryptography

Quantum Probability and Statistics

Quantum Filtering and Control

Quantum Engineering and Metrology P. Shor

State estimation in quantum engineering

Multiparticle entanglement of trapped ions

[Hafner et al, Nature 2005]

Experiment validation: statistical ‘reconstruction’ of the quantum state

48 − 1 = 65 535 parameters to estimate (8 ions)

10 hours measurement time

weeks of computer time (‘maximum likelihood’)

System identification for complex dynamics

Photosynthesis: energy from light is transferred to a reaction center

Complex system in noisy environment

Theoretical modelling in parallel with statistical ‘systemidentification’

Find appropriate preparation and measurement designs

Quantum Filtering and ControlSuppression of spin-projection noise – closed loop

JM Geremia, John Stockton and HM, quant-ph/0401107v4

real-time feedbackenables robust

spin-noise suppression

[Quantum Magnetometer, Mabuchi Lab]

Observe and control quantum systems in real time

Dynamics governed by Quantum Stochastic Differential equations

Need for effective low dimensional dynamical models (e.g. ‘Gaussianapprox.’)

Quantum mechanics as a probability theory

States

Observables

Quantum mechanics as a probabilistic theory

States

Observables

Measurements

Channels

Instruments

Quantum states

Complex Hilbert space of ‘wave functions’ H = Cd , L2(R)...

State = preparation: ‘density matrix’ ρ on H

I ρ = ρ∗ (selfadjoint)I ρ ≥ 0 (positive)I Tr(ρ) = 1 (normalised)

Pure state: one dimensional projection Πψ = |ψ〉〈ψ| with ‖ψ‖ = 1

Mixed state: convex combination of pure states ρ =∑i qiΠψi

Natural distances: τ := ρ1 − ρ2

‖τ‖1 := Tr(|τ |) ‖τ‖22 := Tr(τ 2), h(ρ1, ρ2) := 1−Tr

(√ρ

1/21 ρ2ρ

1/21

)

Example: spin (qubit) states

Any density matrix ρ on C2 is of the form

ρ(~r) :=1

2

(1 + rz rx − iryrx + iry 1− rz

)=

1

2(1+ rxσx + ryσy + rzσz) , ‖~r‖ ≤ 1

ρ(~r) is pure if and only if ‖~r‖ = 1

Bloch sphere representation

z

y

x

!r

Quantum observablesObservable: selfadjoint operator A on H

Spectral Theorem (diagonalisation):

A =

∫

σ(A)⊂RaΠ(da) (A =

∑

j

ajΠj)

Probabilistic interpretation: measuring A gives random outcomeA ∈ aj

Pρ[A = aj ] = pj = Tr(ρΠj)

Quantum and classical expectations

Tr(ρf (A)) =∑

j

f (aj)Tr(ρΠj) =∑

j

f (aj)pj = Eρ(f (A))

Example: spin components

Components of spin in x , y , z directions are given by the Paulimatrices

σx :=

(0 11 0

), σy :=

(0 −ii 0

), σz :=

(1 00 −1

)

Let ρ = 12 (1+~r~σ) then

Pρ[σi = ±1] = (1± ri )/2

Different spin components are incompatible: σxσy − σyσx = 2iσz

Indirect measurements

Most real measurements are

I indirectI extended in time

tion of single-system properties, such as the distributionof the lengths of the periods of strong fluorescence, re-quired some effort, which eventually led to the develop-ment of the quantum-jump approach. Apart from theinteresting theoretical implications for the study of indi-vidual quantum systems, Dehmelt’s proposal obviouslyhas important practical applications. An often-cited ex-ample is the realization of a new time standard using asingle atom in a trap. The key idea here is to use eitherthe instantaneous intensity or the photon statistics of theemitted radiation on the strong transition (the statisticsof the bright and dark periods) to stabilize the frequencyof the laser on the weak transition. This is possible be-cause the photon statistics of the strong radiation de-pends on the detuning of the laser on the weak transi-tion (Kim, 1987; Kim and Knight, 1987; Kim et al., 1987;Ligare, 1988; Wilser, 1991). Therefore a change in thestatistics of bright and dark periods indicates that thefrequency of the weak laser has shifted and has to beadjusted. However, for continuously radiating lasers thisfrequency shift will also depend on the intensity of thelaser on the strong transition. Therefore, in practice,pulsed schemes are preferable for frequency standards(Arecchi et al., 1986; Bergquist et al., 1994).

Due to the inability of experimentalists to store, ma-nipulate, and observe single-quantum systems (ions) atthe time of Dehmelt’s proposal, both the practical andthe theoretical implications of his proposal were not im-mediately investigated. It was about ten years later thatthis situation changed. At that time Cook and Kimble(1985) made the first attempt to analyze the situationdescribed above theoretically. Their advance was stimu-lated by the fact that by that time it had become possibleto actually store single ions in an ion trap (Paul trap;Paul et al., 1958; Neuhauser et al., 1980; Paul, 1990).

In their simplified rate-equation approach Cook andKimble started with the rate equations for an incoher-ently driven three-level system as shown in Fig. 1 and

assumed that the strong 0↔1 transition is driven tosaturation. They consequently simplify their rate equa-tions, introducing the probabilities P1 of being in themetastable state and P2 of being in the strongly fluo-rescing 0↔1 transition. This simplification now allowsthe description of the resonance fluorescence to be re-duced to that of a two-state random telegraph process.Either the atomic population is in the levels 0 and 1, andtherefore the ion is strongly radiating (on), or the popu-lation rests in the metastable level 2, and no fluores-cence is observed (off). They then proceed to calculatethe distributions for the lengths of bright and dark peri-ods and find that their distribution is Poissonian. Theiranalysis, which we have outlined very briefly here, is ofcourse very much simplified in many respects. The mostimportant point is certainly the fact that Cook andKimble assume incoherent driving and therefore adopt arate-equation model. In a real experiment coherent ra-diation from lasers is used. The complications arising incoherent excitation finally led to the development of thequantum-jump approach. Despite these problems, theanalysis of Cook and Kimble showed the possibility ofdirect observation of quantum jumps in the fluorescenceof single ions, a prediction that was confirmed shortlyafterwards in a number of experiments (Bergquist et al.,1986; Nagourney et al., 1986a, 1986b; Sauter et al., 1986a,1986b; Dehmelt, 1987) and triggered a large number ofmore detailed investigations, starting with early worksby Javanainen (1986a, 1986b, 1986c). The subsequent ef-fort of a great number of physicists eventually culmi-nated in the development of the quantum-jump ap-proach. Before we present this development in greaterdetail, we should like to study in slightly more detailhow the dynamics of the system determines the statisticsof bright and dark periods. Again assume a three-levelsystem as shown in Fig. 1. Provided the 0↔1 and 0↔2Rabi frequencies are small compared with the decayrates, one finds for the population in the strongly fluo-rescing level 1 as a function of time something like thebehavior shown in Fig. 3 (we derive this in detail in alater section). We choose for this figure the valuesg1@g2 for the Einstein coefficients of levels 1 and 2,which reflects the metastability of level 2. For timesshort compared with the metastable lifetime g2

21 , theatomic dynamics can hardly be aware of level 2 andevolve as a 0–1 two-level system with the ‘‘steady-state’’population r 11 of the upper level. After a time g2

21 , themetastable state has an effect, and the (ensemble-averaged) population in level 1 reduces to the appropri-ate three-level equilibrium values. The ‘‘hump’’ Dr11shown in Fig. 3 is actually a signature of the telegraphicfluorescence discussed above. To show this, consider afew sequences of bright and dark periods in the tele-graph signal as shown in Fig. 4. The total rate of emis-sion R is proportional to the rate in a bright periodtimes the fraction of the evolution made up of brightperiods. This gives

R5g1 r 11S TL

TL1TDD , (1)

FIG. 2. Recorded resonance fluorescence signal exhibitingquantum jumps from a laser-excited 24Mg+ ion (Thompson,1996). Periods of high photon count rate are interrupted byperiods with negligible count rate (except for an unavoidabledark-count rate).

103M. B. Plenio and P. L. Knight: Quantum-jump approach to dissipative dynamics . . .


Recorded fluorescence signal from 1 ion

3 steps

I couple state ρ with ‘environment’ in state σ: ρ =⇒ ρ⊗ σ

I interaction entangles systems 1 & 2: ρ⊗ σ =⇒ U(ρ⊗ σ)U∗

I measure environment observable A =Pi aiΠi

Pρ[A = ai ] = Tr1&2(U(ρ⊗ σ)U∗ 1⊗ Πi )

= Tr1&2(ρ⊗ σU∗(1⊗ Πi )U) = Tr(ρMi )

Positive, normalised linear map: ρ 7→ pi = Tr(ρMi )

General measurements

Definition

A measurement on H with outcomes in (Ω,Σ) is a linear map

M : T1(H)→ L1(Ω,Σ, µ)

such that pρ := M(ρ) is a probability density w.r.t. µ for each state ρ.

Any M is of the form

Pρ(E ) =

∫

E

pρdµ = Tr(ρm(E ))

for some Positive Operator valued Measure (POVM) m(E ) : E ∈ Σ.

Naimark’s TheoremAny measurement can be realised indirectly by ‘usual’

projection measurement on the environment

Quantum Instrument

! !!!!!!!!!!!!!!

" !!!!!!!!!!!!!!

!!!!!!!!!!!!!! !i

J!

A = ai

Measure B =∑j bjQj and A =

∑i aiPj

P[B = bj &A = ai ] = Tr(U(ρ⊗ σ)U∗ Qj ⊗ Pi )

= Tr(U(ρ⊗ |ψ〉〈ψ|)U∗ Qj ⊗ |ei 〉〈ei |)

= Tr(ViρV∗i Qj ) = piTr(ρiQj )

where Vi := 〈ei ,Uψ〉 are the Kraus operators with∑i V∗i Vi = 1

Conditional state ρi = ViρV∗i /pi

Quantum Channels

! !!!!!!!!!!!!!!

" !!!!!!!!!!!!!!

!!!!!!!!!!!!!! C(!)J

!

DefinitionA quantum channel is a completely positive, normalised linear map

C : T1(H)→ T1(H)

Stinespring-Kraus Theorem

Any channel C is of the form

C (ρ) =∑

i

ViρV∗i ,

for some operators Vi satisfying∑i V∗i Vi = 1.

Any channel can be realised indirectly by ‘usual’ product construction with some

environment

Summary of quantum probability

States are the analogue of probability distributions

Observables are the analogue of random variables

Dualities: B(H) = T1(H)∗ and L∞(Ω,Σ, µ) = L1(Ω,Σ, µ)∗

Measurements are quantum-to-classical randomisations ρ 7→ Pρ

Channels are quantum-to-quantum randomisations ρ 7→ C (ρ)

Instruments are quantum-to-mixed (classical and quantum)randomisations

Quantum Statistics

The 70’s

Some current topics

State estimation

Quantum Statistics in the 70’s

Helstrom, Holevo, Belavkin, Yuen, Kennedy...

Formulated and solved first quantum statistical decision problems

I quantum statistical model Q = ρθ : θ ∈ Θ

I decision problem (estimation, testing)

I find optimal measurement (and estimator)

Quantum Gaussian states, covariant families, state discrimination...

Elements of a (purely) quantum statistical theoryI Quantum Fisher Information

I Quantum Cramer-Rao bound(s)

I Holevo bound (now known to be the asymptotic quantumCramer-Rao bound)

I ...

Asymptotics in state estimation

!! !!!!!!!!!!

!! !!!!!!!!!!

!! !!!!!!!!!!

M(n) ! X(n) ! "n

(Asymptotically) optimal measurements and rates for d = 2

[Gill and Massar, P.R.A. 2002] [Bagan et al. (incl. Gill), P.R.A. 2006]

[Hayashi and Matsumoto 2004] [Gill, 2005]

Local asymptotic normality for d <∞[Guta and Kahn, C.M.P. 2009]

Quantum Homodyne Tomography

I.I.D. samples from Radon transform of the Wigner function[Vogel and H. Risken., P.R.A. 1989]

[Breitenbach et al, Nature 1997]

Estimation of infinite dimensional states (non-parametric)[Artiles Guta and Gill, J.R.S.S. B, 2005] [Butucea, Guta and Artiles, Ann.Stat. 2007]

State estimation and compressed sensing

A0 := 1, A1, . . . ,Ad2−1 basis in M(Cd)

State ρ is characterised by Fourier coefficients ai := Tr(ρAi )

Often ρ is known to be ‘sparse’ (Rank(ρ) = r d)

How many (and which) observables are sufficient to estimate ρ ?

Similar to the matrix completion problem (Netflix)

[Candes and Recht, Found. Comp. Math. 2008]

State estimation and compressed sensing

Theorem [Gross 2009]

A0 = 1,A1, . . . ,Ad2−1 ‘incoherent basis’

Choose i1, . . . , im ∈ 1, . . . , d2 − 1 randomly with m = cdr(log d)2

Measure Aik and obtain the averages ak = Tr(ρAik )

Then with high probability ρ is the unique solution of the s.d.o. problem

L1-minimisation: minimise ‖τ‖1 with constraints

Tr(τ) = 1 and Tr(τAik ) = ak

The proof uses a Bernstein inequality for matrix valued r.v.

[Ahlswede and Winter, IEEE Trans.Inf.Th. 2002]

P

[

‖m∑

i=1

Xi‖ > t

]

≤ 2de−t2/4mσ2

, σ2 = ‖E(X 2)‖

Asymptotics in state discrimination

Two hypotheses ρ⊗n and σ⊗n

Test Mn = A1,n, A2,n = 1− A1,n

Error probabilities

α(Mn) = Tr(ρ⊗n(1− A1,n)), β(Mn) = Tr(σ⊗nA1,n)




Error probabilities


Quantum Stein Lemma

Let βn(ε) = infβ(Mn) : α(Mn) ≤ ε. Then

limn→∞

1

nlog βn(ε) = −S(ρ|σ) := −Tr(ρ(log ρ− log σ))

[Hiai and Petz, C.M.P. 1991] [Ogawa and Nagaoka IEEE Trans. Inform. 2000]




Error probabilities


Quantum Chernoff bound

Let pn = infπ1α(Mn) + π2β(Mn) : Mn for prior (π1, π2). Then

limn→∞

1

nlog pn = − log

(infs∈[0,1]

Tr(ρsσ1−s)

)

[Nussbaum and Szkola, Ann. Stat. 2009] [Audenaert et al C.M.P. 2008]

Estimation of Quantum Channels

ρn

∼∼∼∼∼

∼∼∼∼∼

∼∼∼∼∼

Cθ

Cθ

Cθ

∼∼∼∼∼

∼∼∼∼∼

∼∼∼∼∼

Mn

- X ∼ PθM,n

- θn

Fast(er) estimation rates (n−2) for entangled input states

[Kahn, P.R.A. 2007]

Applications in Quantum Metrology[Giovanetti et al, Science 2004]

Quantum Cloning and Quantum Benchmarks

Quantum no-cloning Theorem[381 papers on arXiv.org]

Canal quantique

ρθ∼∼∼∼∼∼∼∼∼∼∼∼∼∼ C

∼∼∼∼∼∼∼∼∼∼∼∼∼∼ ρθ

∼∼∼∼∼∼∼∼∼∼∼∼∼∼ ρθ

• “Randomisation quantique”

• Transmission par un canal avec du bruit

• Perte de coherence provoquee par l’interaction avec

l’environnement

Measure and prepare scheme vs teleportation[Hammerer et al P.R.L. 2005] [Owari et al N.J.P. 2008]

!! !!!!!!!!!! M ! X ! P(M)! ! P !!!!!!!!!! !X

Quantum state estimation

Set-up

Example: spin rotation model

Quantum Cramer-Rao bound

Quantum Gaussian states

Set-up of quantum estimation problems

Quantum statistical model over Θ:

Q = ρθ : θ ∈ Θ

Estimation procedure: measure state ρθ and devise estimatorθ = θ(R)

!! !!!!!!!!!!!!!!!M R ! P(M)

! ! !

Measurement design:

I which classical model P(M) = P(M)θ : θ ∈ Θ is ‘best’ ?

I trade-off between incompatible observables

I optimal measurement depends on statistical problem

Quantum Cramer-Rao bound(s)*

Theorem [Helstrom, Holevo, Belavkin]

Let Q = ρθ : θ ∈ Rk be a ‘smooth’ quantum model.

For any unbiased measurement M with outcome θ ∈ Rk

Var(θ) ≥ I (M)(θ)−1 ≥ H(θ)−1

Helstrom’s Quantum Fisher information matrix

H(θ)i,j := Tr(ρθLθ,i Lθ,j)

Symmetric logarithmic derivatives: ∂ρθ∂θj

= ρθ Lθ,j

*= several inequivalent C.R. bounds exist depending on symmetrisation

Proof (projection valued measurements)Measure observable X and get result X ≡ θ ∼ Pθ

Hilbert spaces L2(ρθ) and L2(R,Pθ)

〈A,B〉θ := Tr(ρθA B) 〈f (X), g(X)〉θ = Eθ(f (X)g(X))

Fisher informations

H(θ) := Tr(ρθL2θ) = ‖Lθ‖

2θ

I (M)(θ) := Eθ(`2θ(X)) = ‖`θ(X)‖2

θ

Isometry

I : L2(R,Pθ) → L2(ρθ)

f (X) 7→ f (X )

Eθ(f (X)g(X)) = Tr(ρθf (X ) g(X )) = 〈f (X ), g(X )〉θ

Proof for projection valued measurements

The projection of Lθ onto L2(R,Pθ) is `θ(X )

Indeed for every f ∈ L2(pθ)

〈f (X), `θ(X)〉θ =dEθ(f )

dθ= Tr

(dρθdθf (X )

)= 〈f (X ),Lθ〉θ

=⇒ ‖`θ‖2θ ≤ ‖Lθ‖

2θ

Bound achieved (locally) at θ0 by

X = θ01+Lθ0

H(θ0)

Trade-off between parameters

One-dimensional model: C.R. bound can be achieved asymptotically

1. measure fraction n n of systems to obtain rough estimator θ0

2. measure L(n)θ0

:= Lθ0 ⊗ 1⊗ · · · ⊗ 1+ · · ·+ 1⊗ 1⊗ · · · ⊗ Lθ0

3. set θn := θ0 + L(n)θ0/H(θ0)

Multi-dimensional model: H(θ) is achievable iff

Tr(ρθ[Lθ,j , Lθ,i ]) = 0, ∀1 ≤ i , j ≤ k

Trade-off between estimation of different coordinates

Optimal measurement depends on loss function (Holevo bound) andasymptotic risk is not simply expressible in terms of some quantuminformation matrix

Optimal estimation using local asymptotic normality

!! !!!!!!!!!! H ! Y ! P(H,Φθ) ! θ

!! !!!!!!!!!!

!! !!!!!!!!!!

!! !!!!!!!!!!

Mn ! Xn ! P(Mn, !!) ! "n

n ! "

Sequence of I.I.D. quantum statistical models Qn = ρ⊗nθ : θ ∈ Θ

Qn converges (locally) to simpler Gaussian shift model Q

Optimal measurement for limit Q can be pulled back to Qn

Quantum Gaussian states

Quantum ‘particle’ with canonical observables Q,P on H = L2(R)

QP − PQ = i1 (Heisenberg’s commutation relations)

Centred Gaussian state Φ

Tr (Φ exp(−ivQ − iuP)) = exp

(−

1

2

(u v

)V

(uv

))

with ‘covariance matrix’ V satisfying the uncertainty principle

Det(V ) =

∣∣∣∣∣∣

Tr(ΦQ2) Tr(ΦQ P)

Tr(ΦQ P) Tr(ΦP2)

∣∣∣∣∣∣≥

1

4

Examples

Vaccum state |0〉

V = Diag( 12 ,

12 )

P

Q

|0!

Thermal equilibrium state Φ(s)

V = Diag( s2 ,s2 )

P

Q

!(s)

Squeezed state |0, ξ〉

V = Diag( e−ξ

2 , eξ

2 )

P

Q

|0, !!

Quantum Gaussian shift model(s)

Displacement operator D(u, v) := exp(ivQ − iuP)

Coherent (laser) state

|u, v〉 := D(u, v)|0〉

P

Q

v

u

|u, v!

Displaced thermal stateΦ(u, v ; s) = D(u, v)Φ(s)D(u, v)∗

P

Q

v

u

!(u, v ; s)

Optimal measurement for Gaussian shift

Oscillator (Q,P) in state |u, v〉

Oscillator (Q ′,P ′) in vacuum state |0〉

Noisy coordinates commute: [Q+,P−] = 0

Q± := Q ± Q ′

P± := P ± P ′

(Q,P )

(Q!, P !)

(Q+, P+)

(Q!, P!)

Heterodyne measurement (Q+,P−) gives estimator(u, v) ∼ N((u, v), 1)

TheoremThe heterodyne measurement is optimal among covariant measurementsand

achieves the minimax risk for the loss function |u − u|2 + |v − v |2.

Outlook

Statistics is more and more employed in quantum experiments

Remarkable coherence between quantum and classical statistics

Trade-off between estimation of different parameters

Optimal measurement depends on decision problem

More information:

Madalin Guta’s Quantum Statistics course

http://maths.dept.shef.ac.uk/magic/course.php?id=64

Madalin Guta’s Lunteren lectures

http://www.maths.nottingham.ac.uk/personal/pmzmig/Lunteren.pdf

References

R.D. Gill (2001)

Teleportation into quantum statistics

J. Korean Statistical Society 30, 291–325

arXiv:math.ST/0405572

R.D. Gill (2001)

Asymptotics in quantum statistics

IMS Monographs 36, 255–285

arXiv:math.ST/0405571

R.D. Gill (2005)

Asymptotic information bounds in quantum statistics.

QP and PQ

arXiv:math.ST/0512443.

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